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The remainder when $5^{99}$ is divided by $13$ is

$\begin{array}{1 1} 6 \\ 7 \\ 8 \\ 9 \end{array}$

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$5^{99}=5.(5^2)^{49}$
$=5.(26-1)^{49}$
$=5\big[^{49}C_0.(26)^{49}-^{49}C_1.(26)^{48}+^{49}C_2(26)^{47}.-.....-^{49}C_{49}\big]$
$5(26m-1)$ where $m\in Z$
$=(5\times 26m-5+13-13)$
$=13(10m-1)+8$
$\therefore 5^{99}$ leaves remainder $8$ when divided by $13$
answered Oct 26, 2013 by rvidyagovindarajan_1
 
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